Select the application file in the Finder and then Get Info on it by either selecting "Get Info" in the File menu, or hitting cmd-I. In the info window, check the, "Open in 32 Bit Mode" checkbox and close the window. Now it should work.

There is an extremely simple fix for this problem that works flawlessly on my Mac Pro (64-bit):

With a text editor open the file:Arduino 16.app/Contents/Info.plist

Edit the lines: <key>JVMVersion</key> <string>1.4+</string>

by removing the + after the 1.4. What you are telling Mac OS is that you want the app to use the 1.4 version of Java, and no other. Save and close the file and pronto, the Arduino 16.app will start up no problem.